Simulation of Markov Chains with Continuous State Space by Using Simple Stratified and Sudoku Latin Square Sampling

Abstract

Monte Carlo (MC) is widely used for simulating discrete time Markov chains. Here, N copies of the chain are simulated in parallel, using pseudorandom numbers. We restrict ourselves to a one-dimensional continuous state space. We analyze the effect of replacing pseudorandom numbers on $I := [0,1)$ with stratified random points over $I^2$: for each point, the first component is used to select a state and the second component is used to advance the chain by one step. Two stratified sampling techniques are compared: simple stratified sampling (SSS) and Sudoku Latin square sampling (SLSS). For both methods and for $N=p^2$ samples, the unit square is dissected into $p^2$ subsquares and there is one sample in each subsquare. For SLSS, each side of the unit square is divided into N subintervals and the projections of the samples on the side are distributed with one projection in each subinterval. Stratified strategies outperform classical MC if the N copies are reordered by increasing states at each step. We prove that the variance of SSS and SLSS estimators is bounded by $\mathcal {O}(N^{-3/2})$, while it is bounded by $\mathcal {O}(N^{-1})$ for MC. The results of numerical experiments indicate that these upper bounds match the observed rates. They also show that SLSS gives a smaller variance than SSS.